How do I convert an svg (or a plane) into a rounded mesh?
The manual way of doing this would probably be to extrude it up and then use the"Shrink/Fatten" (Alt+S) tool which scales each Vertex/Edge/Face along its normal which should give you something you might be able to work with.

I wouldn't be surprised if there was already a tool or an add-on to do this more or less automatically but sadly I don't know it